Open Access

Iterative process for a strictly pseudo-contractive mapping in uniformly convex Banach spaces

Journal of Inequalities and Applications20142014:377

https://doi.org/10.1186/1029-242X-2014-377

Received: 14 February 2014

Accepted: 7 August 2014

Published: 29 September 2014

Abstract

This paper is concerned with a new method to prove the weak convergence of a strictly pseudo-contractive mapping in a p-uniformly convex Banach space with more relaxed restrictions on the parameters. Our results extend and improve the corresponding earlier results.

MSC:41A65, 47H17, 47J20.

Keywords

strictly pseudo-contractive mappingnonexpansive mappingMann iterationuniformly convex Banach space

1 Introduction and preliminaries

In 1967, Browder and Petryshyn [1] gave the classical definition for strictly pseudo-contractive mappings in Hilbert spaces for the first time.

Definition 1.1 Let C be a nonempty closed convex subset of a real Hilbert space H. T : C H is called a Browder-Petryshyn-type k-strictly pseudo-contractive mapping. Then there exists k [ 0 , 1 ) such that for every x , y C
T x T y , j ( x y ) x y 2 k ( I T ) x ( I T ) y 2 .
(1.1)

In 2010, Zhou [2] gave a new definition for k-strictly pseudo-contractive mappings in q-uniformly smooth Banach spaces.

Definition 1.2 Let C be a nonempty closed convex subset of a q-uniformly smooth Banach space X. T : C C is called a Zhou-type k-strictly pseudo-contractive mapping, if there exists k [ 0 , 1 ) such that for every x , y C
T x T y , j q ( x y ) x y q 1 k 2 ( I T ) x ( I T ) y q .
(1.2)

In 2009, Hu and Wang [3] gave another definition for k-strictly pseudo-contractive mappings in p-uniformly convex Banach spaces.

Definition 1.3 Let C be a nonempty closed convex subset of a p-uniformly convex Banach space X. T : C C is called a Hu-type k-strictly pseudo-contractive mapping, if there exists k [ 0 , 1 ) such that for every x , y C
T x T y p x y p + k ( I T ) x ( I T ) y p .
(1.3)

Remark 1.1 The mappings defined by (1.1) and (1.2) are pseudo-contractive mappings, but the mapping defined by (1.3) may not be pseudo-contractive in general Banach spaces.

Remark 1.2 If and only if q = 2 , the mappings defined by (1.1) and (1.2) are equivalent.

Remark 1.3 If p = q = 2 , the mappings defined by (1.1), (1.2), and (1.3) are equivalent in Hilbert space.

In 1979, Reich [4] established a weak convergence theorem via a Mann-type iterative process for nonexpansive mapping in a uniformly convex Banach space with Fréchet differentiable norm.

Theorem R Let C be a closed convex subset of a uniformly convex Banach space X with a Fréchet differentiable norm and T : C C a nonexpansive mapping with F ( T ) . For any x 1 C , the iterative sequence { x n } is defined by x n + 1 = ( 1 α n ) x n + α n T x n , where the real sequence { α n } [ 0 , 1 ] and n = 1 ( 1 α n ) α n = . Then the sequence { x n } converges weakly to a fixed point of T.

In 2007, Marino and Xu [5] improved Reich’s [4] result and gave several weak convergence theorems via the normal Mann iterative algorithm for strictly pseudo-contractive mappings in Hilbert spaces. Further, they proposed an open problem: Do the main results of [5]still hold true in the framework of Banach spaces which are uniformly convex and have a Fréchet differentiable norm?

In 2009, Hu and Wang [3] considered above problem in a p-uniformly convex Banach space and established the following theorem.

Theorem H Let C be a closed convex subset of a p-uniformly convex Banach space X with a Fréchet differentiable norm and T : C C be a k-strictly pseudo-contractive mapping in the light of (1.3) with coefficients p , k < min { 1 , 2 ( p 2 ) c p } and F ( T ) . For any x 1 C and n > 1 , the iterative sequence { x n } is defined by x n + 1 = ( 1 α n ) x n + α n T x n , where the real sequence { α n } [ 0 , 1 ] and 0 < ε α n 1 ε < 1 2 p 2 k c p . Then the sequence { x n } converges weakly to a fixed point of T.

Question Can one relax the restriction on the parameters α n in Theorem H and simplify its proof?

The purpose of this paper is to solve the question mentioned above. To prove our results, we need the following lemmas.

Lemma 1.1 (see [3])

Let C be a nonempty closed convex subset of a p-uniformly convex Banach space X and T : C C be a Hu-type strictly pseudo-contractive mapping in the light of (1.3). For α ( 0 , 1 ) , define T α : C C by T α = ( 1 α ) x + α T x , for x C . If α ( 0 , 1 ( k 2 p 2 ) / c p ) , then T α is a nonexpansive mapping and F ( T α ) = F ( T ) .

Lemma 1.2 (see [3])

Let C be a nonempty closed convex subset of a p-uniformly convex Banach space X and T : C C be a Hu-type strictly pseudo-contractive mapping in the light of (1.3). For μ ( 0 , 1 ) , T μ : C C is defined by T μ = ( 1 μ ) x + μ T x , for x C . Then the following inequality holds:
T μ x T μ y p x y p ( W p ( μ ) c p μ λ ) ( I T ) x ( I T ) y p , x , y C ,

where W p ( μ ) = μ p ( 1 μ ) + μ ( 1 μ ) p .

Lemma 1.3 (see [6])

Let C be a nonempty closed convex subset of a p-uniformly convex Banach space X and T : C C be a nonexpansive mapping, then I T is demiclosed at zero.

Lemma 1.4 (see [7])

Let C be a nonempty closed convex subset of a p-uniformly convex Banach space X which satisfies the Opial condition and T : C C be a quasi-nonexpansive mapping with F ( T ) . If I T is demiclosed at zero, then for any x 0 C , the normal Mann iteration { x n } defined by
x n + 1 = ( 1 α n ) x n + α n T x n , n 0 ,

converges weakly to a fixed point of T, where { α n } [ 0 , 1 ] and n = 0 min { α n , ( 1 α n ) } = .

Lemma 1.5 (see [7])

Let C be a nonempty closed convex subset of a p-uniformly convex Banach space X whose dual space X satisfies Kadec-Klee property and T : C C be a nonexpansive mapping with F ( T ) . Then, for any x 0 C , the normal Mann iteration { x n } defined by
x n + 1 = ( 1 α n ) x n + α n T x n , n 0 ,

converges weakly to a fixed point of T, where { α n } [ 0 , 1 ] and n = 0 min { α n , ( 1 α n ) } = .

Now we are in a position to state and prove the main results in this paper.

2 Main results

Theorem 2.1 Let C be a nonempty closed convex subset of a p-uniformly convex Banach space X with Fréchet differential norm. Let T : C C be a Hu-type k-strictly pseudo-contractive mapping in the light of (1.3) with coefficients p , k < min { 1 , 2 ( p 2 ) c n } and F ( T ) . Assume that a real sequence { α n } in [ 0 , 1 ] satisfies the conditions:
  1. (i)

    0 α n α = 1 ( k 2 p 2 / c p ) , n 0 ;

     
  2. (ii)

    n = 0 α n [ ( 1 α n ) 2 2 p c p k ] = .

     
For any x 0 C , the normal Mann iterative sequence { x n } is defined by
x n + 1 = ( 1 α n ) x n + α n T x n , n 0 .
(2.1)

Then the sequence { x n } defined by (2.1) converges weakly to a fixed point of T.

Proof Let T α be given as in Lemma 1.1. Then T α : C C is a nonexpansive mapping with F ( T α ) = F ( T ) . Set β n = α α n α . Then (2.1) reduces to x n + 1 = β n x n + ( 1 β n ) T α x n .

We note that
n = 0 β n ( 1 β n ) = 1 α 2 n = 0 α n ( α α n ) = 1 α 2 n = 0 α n ( 1 α n k 2 p 2 c p ) = 2 p 2 α 2 c p n = 0 α n [ ( 1 α n ) 2 p 2 c p k ] = .

By using Theorem R, we conclude that { x n } converges weakly to a fixed point of T α , and of T. The proof is complete. □

Remark 2.2 Theorem 2.1 relaxes the iterative parameters in Theorem H and our proof method is also quite concise.

Theorem 2.3 Let C be a nonempty closed convex subset of a p-uniformly convex Banach space X which satisfies the Opial condition. Let T : C C be a Hu-type k-strictly pseudo-contractive mapping in the light of (1.3) with coefficients p , k < min { 1 , 2 ( p 2 ) c p } and F ( T ) . Assume that the real sequence { α n } in [ 0 , 1 ] satisfies the conditions:
  1. (i)

    0 α n α = 1 ( k 2 p 2 / c p ) , n 0 ;

     
  2. (ii)

    n = 0 α n [ ( 1 α n ) 2 2 p c p k ] = .

     
For any x 0 C , the normal Mann iteration { x n } is defined by
x n + 1 = ( 1 α n ) x n + α n T x n , n 0 .
(2.2)

Then the sequence { x n } defined by (2.2) converges weakly to the fixed point of T.

Proof Let T α be given as in Lemma 1.1. Then T α : C C is a nonexpansive mapping with F ( T α ) = F ( T ) . Set β n = α α n α . Then (2.2) reduces to x n + 1 = β n x n + ( 1 β n ) T α x n . As shown in Theorem 2.1, n = 0 β n ( 1 β n ) = . By Lemma 1.3, I T α is demiclosed at zero. By Lemma 1.4, we conclude that { x n } converges weakly to a fixed point of T α , and of T. The proof is complete. □

Theorem 2.4 Let C be a nonempty closed convex subset of a p-uniformly convex Banach space X with the dual space X satisfying the Kadec-Klee property. Let T : C C be a Hu-type k-strictly pseudo-contractive mapping in the light of (1.3) with coefficients p , k < min { 1 , 2 ( p 2 ) c p } and F ( T ) . Assume that the real sequence { α n } in [ 0 , 1 ] satisfies the conditions:
  1. (i)

    0 α n α = 1 ( k 2 p 2 / c p ) , n 0 ;

     
  2. (ii)

    n = 0 α n [ ( 1 α n ) 2 2 p c p k ] = .

     
For any x 0 C , the normal Mann iteration { x n } is defined by
x n + 1 = ( 1 α n ) x n + α n T x n , n 0 .
(2.3)

Then the sequence { x n } defined by (2.3) converges weakly to a fixed point of T.

Proof Let T α be given as in Lemma 1.1. Then T α : C C is a nonexpansive mapping with F ( T α ) = F ( T ) . Set β n = α α n α . Then (2.3) reduces to x n + 1 = β n x n + ( 1 β n ) T α x n . As shown in Theorem 2.1, n = 0 β n ( 1 β n ) = . By using Lemma 1.5, { x n } defined by (2.3) converges weakly to a fixed point of T α , and of T. The proof is complete. □

Declarations

Acknowledgements

This research was supported by the National Natural Science Foundation of China (11071053).

Authors’ Affiliations

(1)
Department of Mathematics, Shijiazhuang Mechanical Engineering College
(2)
Department of Mathematics and Information, Hebei Normal University

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© Zhou et al.; licensee Springer. 2014

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